As the swim coach at Covington, Umaima selects which athletes will participate in the state-wide swim relay. The relay team swims $\frac{3}{4}$ of a mile in total, with each team member responsible for swimming $\frac{3}{8}$ of a mile. The team must complete the swim in $\frac{3}{4}$ of an hour. How many swimmers does Umaima need on the relay team?
Answer: To find out how many swimmers Umaima needs on the team, divide the total distance ( $\frac{3}{4}$ of a mile) by the distance each team member will swim ( $\frac{3}{8}$ of a mile). $ \dfrac{{\dfrac{3}{4} \text{ mile}}} {{\dfrac{3}{8} \text{ mile per swimmer}}} = {\text{ number of swimmers}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{3}{8} \text{ mile per swimmer}}$ is ${\dfrac{8}{3} \text{ swimmers per mile}}$ $ {\dfrac{3}{4}\text{ mile}} \times {\dfrac{8}{3} \text{ swimmers per mile}} = {\text{ number of swimmers}} $ $ \dfrac{{3} \cdot {8}} {{4} \cdot {3}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $3$ in the numerator and the $3$ in the denominator by $3$ $ \dfrac{{\cancel{3}^{1}} \cdot {8}} {{4} \cdot {\cancel{3}^{1}}} = {\text{ number of swimmers}} $ Reduce terms with common factors by dividing the $8$ in the numerator and the $4$ in the denominator by $4$ $ \dfrac{{1} \cdot {\cancel{8}^{2}}} {{\cancel{4}^{1}} \cdot {1}} = {\text{ number of swimmers}} $ Simplify: $ \dfrac{{1} \cdot {2}} {{1} \cdot {1}} = {2} $ Umaima needs 2 swimmers on her team.